AI 中文总结
本文将量子TDA作为实用特征提取方法，开发矩基量子算法，经实验验证其可从高维数据中提取拓扑特征，在时间序列分析中提升预测性能，突破经典TDA的成本瓶颈。
AI 中文摘要
拓扑数据分析（TDA）为从复杂、非结构化数据的形状中提取信息提供了强大框架，但计算高维拓扑特征的经典成本限制了其应用。量子TDA算法为突破这一瓶颈提供了途径，不过现有方法通常聚焦于精确或高精度的贝蒂数估计，使得实用量子优势的适用范围显得较窄。本文转而将量子TDA构建为下游数据分析的特征提取方法，通过从组合拉普拉斯算子中提取低阶谱信息作为高维拓扑的代理，我们从应用和算法两方面为这一视角提供支撑。首先，我们在两项时间序列应用中证实，高阶TDA特征可提升预测性能：用于神经退行性疾病分类的功能磁共振成像分析，以及用于识别市场不稳定性的金融时间序列分析。其次，我们开发了一种基于矩的量子算法，表明低阶矩（包括相对迹）与高维贝蒂信息存在强相关性，即便相对贝蒂数很小亦是如此。最后，我们给出了电路构造、资源估计、量子-经典交叉投影，以及来自类IonQ Temo系列的钡基开发系统的实验结果，从图实例中提取拉普拉斯衍生的可观测量，并将其与精确贝蒂信息进行定量比较。这些结果共同确立了量子TDA为从经典处理具有挑战性的数据中提取拓扑特征的实用方法。
英文摘要
Topological data analysis (TDA) provides a powerful framework for extracting information about the shape of complex, unstructured data, but the classical cost of computing high dimensional topological features limits its application. Quantum algorithms for TDA offer a route around this bottleneck, yet existing approaches typically focus on exact or high precision Betti number estimation, making the regime for practical quantum advantage appear narrow. Here, we instead frame quantum TDA as a feature-extraction method for downstream data analysis by extracting low-order spectral information from the combinatorial Laplacian as a proxy for high-dimensional topology. We support this perspective from both the application and algorithmic sides. First, we show that higher-order TDA features improve predictive performance in two time-series applications: functional MRI analysis for neurodegenerative disease classification and financial time-series analysis for identifying market instability. Second, we develop a moment-based quantum algorithm and show that low-order moments, including the relative trace, are strongly correlated with high-dimensional Betti information, even when the relative Betti number is small. Finally, we present circuit constructions, resource estimates, quantum-classical crossover projections, and experimental results from a Barium development system similar to the forthcoming IonQ Tempo line, extracting Laplacian-derived observables from graph instances and quantitatively comparing them with exact Betti information. Together, these results establish quantum TDA as a practical approach for extracting topological features from classically challenging data
Comments38 pages, 24 figures
